Algorithms for Constructing Isolating Sets of Phase Flows and Computer-Assisted Proofs with the Use of Interval Taylor Models

Abstract

This work continues the research devoted to considering interval Taylor models (TM) as applied to proving the existence of periodic trajectories in systems of ordinary differential equations (ODEs). The Taylor models are used here within a topological approach to constructing an isolating set for the ODE phase flow. We prove auxiliary assertions and propose constructive algorithms for validated numerics over TMs with the aim to expand the domain of their applicability. Necessary topological assertions are proved in order to establish the properties of isolating sets constructed with the aid of TMs. As a result, constructive algorithms are formulated and the main theorem is proved. This theorem makes it possible to construct and verify the homotopy equivalence of the isolating set and the one-dimensional sphere and the homotopy of the mapping of the isolating set into itself to the identity mapping for given systems of ODEs. We also prove the computational complexity of the main algorithm and provide an example of its usage.